Tony Blair and Sir John Major have said that if Britain left the EU, border control would be introduced between Northern Ireland and the Republic and the union with Scotland would be threatened.

In a symbolic joint appearance in Derry, the former prime ministers, both of whom played key roles in the Northern Ireland peace process, stressed the risks of voting to leave the EU in the 23 June referendum.

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Speaking to an audience at Ulster University, Major said Brexit would probably lead to a new vote on Scottish independence, while Blair said it would result in the border between Northern Ireland and the Republic becoming the frontier between the UK and the rest of the EU. Major said Brexit would probably lead to a new vote on Scottish independence.

That would make Vote Leave’s pledge to restrict EU immigration impossible to achieve without ditching the common travel area that currently allows the public to move freely between the two countries, he said. “What you would actually have to do is end up with having border controls and customs checks.”

Arron Banks, the funder of the unofficial exit campaign Leave.EU, said he was considering a challenge to the government’s decision to extend the deadline for registering to vote in the EU referendum by 48 hours.

Major and Blair share a platform at the remain campaign event at Ulster University in Derry. Photograph: Reuters

Major, who was Blair’s predecessor in No 10, said on Thursday that leaving the EU would put the Republic of Ireland on “the opposite side of the table”, as the UK sought to negotiate an exit with the 27 other member states.

He also said pressure for a fresh referendum on Scottish independence could become “politically irresistible” if people in Scotland voted to remain in the EU but the UK as a whole pulled out. For that reason, he said, Brexit could undermine the entire constitutional settlement of the UK.

“The plain, uncomfortable truth is that the unity of the UK itself is on the ballot paper in two weeks’ time,” he said. “There is a serious risk of a new referendum, not immediately perhaps, but eventually.”

The two men, who referred to to each other as John and Tony, also underlined the importance of EU membership for employment and business.

Major said: “Just like prime ministers before us and after us, Tony and I fought hard to persuade our fellow EU leaders to continue to build the single market. We did so because it’s of great benefit to Britain.”

‘Against the grain of the future’

Major fought a bitter battle against backbench Tory rebels over Europe during his premiership, and has made a series of spiky interventions in the referendum, including branding one of the leading Vote Leave campaigners, Boris Johnson, a “court jester”.

Blair said that for Britain to split from its neighbours would be to “go against the grain of the future”, which was about countries coming together.

The Northern Ireland secretary, Theresa Villiers, who is campaigning for Britain to leave the EU, said she did not accept that Brexit would end the open-border arrangement. In a statement issued by Vote Leave, she said: “The common travel area between the UK and Ireland has existed for nearly 100 years, since the creation of the Irish state in the 1920s. It will continue if we vote to leave.

“There would be risks to manage but they are not significantly more serious than risks that are already managed effectively today through bilateral cooperation between the UK and Ireland.”